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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zsptri (f07qw)

## Purpose

nag_lapack_zsptri (f07qw) computes the inverse of a complex symmetric matrix A$A$, where A$A$ has been factorized by nag_lapack_zsptrf (f07qr), using packed storage.

## Syntax

[ap, info] = f07qw(uplo, ap, ipiv, 'n', n)
[ap, info] = nag_lapack_zsptri(uplo, ap, ipiv, 'n', n)

## Description

nag_lapack_zsptri (f07qw) is used to compute the inverse of a complex symmetric matrix A$A$, the function must be preceded by a call to nag_lapack_zsptrf (f07qr), which computes the Bunch–Kaufman factorization of A$A$, using packed storage.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving UTPTXPU = D1${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving LTPTXPL = D1${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$.

## References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization of A$A$ stored in packed form, as returned by nag_lapack_zsptrf (f07qr).
3:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the interchanges and the block structure of D$D$, as returned by nag_lapack_zsptrf (f07qr).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

work

### Output Parameters

1:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization stores the n$n$ by n$n$ matrix A1${A}^{-1}$.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A1${A}^{-1}$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A1${A}^{-1}$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: ipiv, 5: work, 6: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, d(i,i)$d\left(i,i\right)$ is exactly zero; D$D$ is singular and the inverse of A$A$ cannot be computed.

## Accuracy

The computed inverse X$X$ satisfies a bound of the form
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, |DUTPTXPUI|c(n)ε(|D||UT|PT|X|P|U| + |D||D1|)$|D{U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU-I|\le c\left(n\right)\epsilon \left(|D||{U}^{\mathrm{T}}|{P}^{\mathrm{T}}|X|P|U|+|D||{D}^{-1}|\right)$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |DLTPTXPLI|c(n)ε(|D||LT|PT|X|P|L| + |D||D1|)$|D{L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL-I|\le c\left(n\right)\epsilon \left(|D||{L}^{\mathrm{T}}|{P}^{\mathrm{T}}|X|P|L|+|D||{D}^{-1}|\right)$,
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision

The total number of real floating point operations is approximately (8/3)n3$\frac{8}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dsptri (f07pj).

## Example

```function nag_lapack_zsptri_example
uplo = 'L';
ap = [ -0.39 - 0.71i;
-7.86 - 2.96i;
0.5278724801640799 - 0.3714660014825906i;
0.442558238872675 + 0.1936483698297402i;
-2.83 - 0.03i;
-0.6078391056683192 + 0.281079647893122i;
-0.4822822975185383 + 0.01498936219105284i;
4.407906236731014 + 5.399120676796941i;
-0.1070821880092683 - 0.3156780862488454i;
-2.095414887840057 - 2.201139281440786i];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = nag_lapack_zsptri(uplo, ap, ipiv)
```
```

apOut =

-0.1562 - 0.1014i
0.0400 + 0.1527i
0.0550 + 0.0845i
0.2162 - 0.0742i
0.0946 - 0.1475i
-0.0326 - 0.1370i
-0.0995 - 0.0461i
-0.1320 - 0.0102i
-0.1793 + 0.1183i
-0.2269 + 0.2383i

info =

0

```
```function f07qw_example
uplo = 'L';
ap = [ -0.39 - 0.71i;
-7.86 - 2.96i;
0.5278724801640799 - 0.3714660014825906i;
0.442558238872675 + 0.1936483698297402i;
-2.83 - 0.03i;
-0.6078391056683192 + 0.281079647893122i;
-0.4822822975185383 + 0.01498936219105284i;
4.407906236731014 + 5.399120676796941i;
-0.1070821880092683 - 0.3156780862488454i;
-2.095414887840057 - 2.201139281440786i];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = f07qw(uplo, ap, ipiv)
```
```

apOut =

-0.1562 - 0.1014i
0.0400 + 0.1527i
0.0550 + 0.0845i
0.2162 - 0.0742i
0.0946 - 0.1475i
-0.0326 - 0.1370i
-0.0995 - 0.0461i
-0.1320 - 0.0102i
-0.1793 + 0.1183i
-0.2269 + 0.2383i

info =

0

```